.
And another, much shorter, review by Patrick Lambelet of the University of California, Santa Barbara:

Beckwith sets out to challenge traditional claims about the origins and development of Buddhism and to show that there was far more fluidity and cross-influence in relations between Indic, Greek, Central Asian, and Chinese cultures than scholars have generally assumed. The project is ambitious, and Beckwith makes a number of intriguing claims, but in taking on so much material across such a wide array of fields, the book comes across as overly confident in its assertions, and sometimes superficial in its treatment.

Among the more provocative claims are that Pyrrho’s philosophy was largely derived from early Buddhist thought; that the Upaniṣadic teachings (and Jainism) did not exist prior to the advent of Buddhism; that the Buddha was of Scythian, and not Indian, origin (his name being Śākamuni, “Sage of the Scythians,” rather than Śākyamuni, the “Sage of the Śākyas”); and that Buddhism may have been directly influenced by Chinese and Central Asian thought. We even have an appendix that asks if Buddhism and Pyrrhonism are both Greek in origin. Perhaps the most compelling argument is the demonstration of similarities between Pyrrho’s philosophical teachings—which had little in common with other contemporaneous Greek schools—and early Buddhist notions of the trilakṣaṇa, or three characteristics (impermanence, unsatisfactoriness, and no-self). However, arguments based on etymological speculations, such as the claim that the name Gautama may have been a derivation of the name Lao Tzu, are less convincing. The sheer scope of the project—encompassing philosophy, philology, history, and archaeology across a broad cultural range—is often disorienting, and ends up overreaching, even if many of the questions are well worth exploring in more depth. For scholars interested in the cultural intersections of philosophy and religion, the book suggests fascinating possibilities, but leaves the reader wishing for more focus and clarity.

The primary piece of evidence that Beckwith provides for the Buddha being
Scythian is that the epithet Śakamuni, in its Gāndhārī spelling, could be inter-
preted to mean ‘Sage of the Sakas’, i.e. an eastern branch of the Scythian people,
rather than the usual ‘Sage of the Sakyans’, i.e. a clan that lived on the border of
present day Nepal and India (a tradition ‘full of chronological and other insuper-
able problems’). From this speculation, we jump to the confident assertion that
‘the Buddha is the only Indian holy man before early modern times who bears
an epithet explicitly identifying him as a non-Indian, a foreigner’ (39)

-Stephen Batchelor's review of "Greek Buddha".

Some well-known indologists seem to like this śākya = śaka speculation:

Śākya is derived from Śaka, one of the principal names of Iranian steppe nomads. Its association with the name Paṇḍu [="white, pale"] is an additional hint of the Iranian origin of the Pāṇḍavas.

-Asko Parpola, 2015, The Roots of Hinduism: The Early Aryans and the Indus Civilization

2. The Sakya have a number of Iranoid links and customs. Briefly:

The name of the Buddha's clan, the Sakya (Skt. s'aakya), cannot be
separated from the designation of the northern Iranian Saka (Skt.
S'aka) hat entered India only after c. 140 BE, via Sistan.

The name, as well as that of some Late Vedic kings and noblemen,
Balhika Pratipiiya and perhaps Cakra Sthapati, recall the Iranian
countries Baaxdhii/Balh (Bactria) and Caxra.

Further, the Sakya, Malla, etc. built high grave mounds, such as the
one for the Buddha. These remind of Central Asian grave mounds (kurgan).

[...etc.]

Taken together, these points tend to indicate that there was some
Iranian influence in Bihar in Late Vedic times. However, by the time
of the Buddha, the Iranoid character of the Sakya, by and large,
seemed to have dissipated. They appeared just like any other eastern
oligarchic tribe and actually claimed descent from the OkkAAka
(IkSvaaku) kings of Ayodhya.

If this śākya = śaka speculation is correct, keeping it for a clan name is quite natural:

"Many East Iranian tribes named themselves Arya ‘noble’, as the Vedic Aryas
and the Persian-Achaemenids. This nomination might not have reflected the self-
consciousness of a gigantic Aryan unity, but rather it must have been a relic of
the common origin of the tribes that had emerged from this unity. Saka, the other
self-designation of East Iranians, means ‘the strong’ (Bailey 1958: 133)."

"Pastoral tribes were very mobile and it is therefore impossible to outline the
borders of their territory. But the ethnic consciousness of ancient kinship is
reflected not only in the self-definition common for all Indo-Iranians – Arya, but
also in the general term Saka which was used by Strabo to describe the region of
Sakasena in West Iran."

Beckwith is aware that in his understanding of early Buddhism he differs from most scholars. Most scholars think Buddhism arose in the same region as Jainism and Ajivikism, approximately at the same time. Buddhism and Jainism (and to some extent Ajivikism) are generally looked upon as responses to a shared problem, linked to the belief in rebirth and karmic retribution. For Beckwith, on the other hand, it is important to show that Buddhism, Jainism and Ajivikism did not originate in the same region at the same time, and he goes to great lengths trying to prove this. Consider what he says about Jainism (p. 160):

In the Jainist view, Mahāvīra, the founder of the Jains, was a contemporary of the Buddha, and they knew each other, as did the founders of the other great ancient Indian sects. This is a wonderful story … Nevertheless, scholars have demonstrated, piece by piece, in many studies of individual contradictory problems in Early Buddhism and other ancient Indian belief systems, that the other traditions have reconfigured themselves so as to be as old as Buddhism, or in some cases, as with the Jains, even older. They demonstrate as clearly as anyone could have done that the story of the Buddha is the oldest of the lot. Although some may wonder why the non-Buddhists made such claims, there are … many good reasons for them to want to imitate the Buddhists’ success.

This passage contains a rather elementary mistake. It describes the situation the wrong way round: the claim that the Buddha and Mahāvīra were contemporaries is not made in Jaina, but in Buddhist texts. This, one would think, should have an effect on Beckwith’s conclusions. In spite of this, he states …

Thanks to the above posters for the interesting links to reviews and critical analysis of Beckwith's book, Greek Buddha. Granted, that many historian/scholars have a message or concept that they are trying to sell. Beckwith was 'brave' enough to put forth his imaginative views based on scanty material that can't possibly be verified through historical records. Without debating the age of Jainism vs Buddhism, etc., there still remains two interesting possibilities that Beckwith touches upon, that the historical Buddha was of Scythian descent, as was the tetralemma that Pyrrho based his philosophical school on. Beckwith seems to think that the tetralemma was a feature of Scythian philosophy that influenced Greek thinkers. If indeed, Buddha was of Scythian origin, the tetralemma could indeed be part of his cultural background.

All of this is speculation, of course. Even if the above was proven to be true, it would change nothing but the historical records and the toilets would still need cleaning!

The Buddha, being from Nepal, was of the Mongoloid race. The Kirats were the ruling people of ancient Nepal:

Kirata Kingdom (Kirat) in Sanskrit literature and Hindu mythology refers to any kingdom of the Kirata people, who were dwellers mostly in the Himalayas (mostly eastern Himalaya). They took part in the Kurukshetra War along with Parvatas (mountaineers) and other Himalayan tribes. They were widespread in the folds and valleys of Himalayas in Nepal and Bhutan, and also migrated to Indian states of Himachal Pradesh, Uttarakhand, Uttar Pradesh, Bihar, West Bengal, Darjeeling (West Bengal), Assam and Tripura including west mountain of Pakistan. Kirata dynasty was established by king Yalamber...https://en.wikipedia.org/wiki/Kirata_Kingdom

This is what the Buddha would look like if he were alive today:

Blue eyed Nepali Sadhu.

"He, the Blessed One, is indeed the Noble Lord, the Perfectly Enlightened One;
He is impeccable in conduct and understanding, the Serene One, the Knower of the Worlds;
He trains perfectly those who wish to be trained; he is Teacher of gods and men; he is Awake and Holy. "
--------------------------------------------"The Dhamma is well-expounded by the Blessed One,
Apparent here and now, timeless, encouraging investigation,
Leading to liberation, to be experienced individually by the wise. "

The Buddha, being from Nepal, was of the Mongoloid race. The Kirats were the ruling people of ancient Nepal:

Kirata Kingdom (Kirat) in Sanskrit literature and Hindu mythology refers to any kingdom of the Kirata people, who were dwellers mostly in the Himalayas (mostly eastern Himalaya). They took part in the Kurukshetra War along with Parvatas (mountaineers) and other Himalayan tribes. They were widespread in the folds and valleys of Himalayas in Nepal and Bhutan, and also migrated to Indian states of Himachal Pradesh, Uttarakhand, Uttar Pradesh, Bihar, West Bengal, Darjeeling (West Bengal), Assam and Tripura including west mountain of Pakistan. Kirata dynasty was established by king Yalamber...https://en.wikipedia.org/wiki/Kirata_Kingdom

Are the Buddha's eyes actually blue, dark-blue or black, which is in fact the same Pali word ‘niila’?

Thomas

I don't know, but I assume the Buddha would have looked like these Indians.

"He, the Blessed One, is indeed the Noble Lord, the Perfectly Enlightened One;
He is impeccable in conduct and understanding, the Serene One, the Knower of the Worlds;
He trains perfectly those who wish to be trained; he is Teacher of gods and men; he is Awake and Holy. "
--------------------------------------------"The Dhamma is well-expounded by the Blessed One,
Apparent here and now, timeless, encouraging investigation,
Leading to liberation, to be experienced individually by the wise. "

Kusala wrote:I did some research on the subject and concluded that the Buddha, without a shadow of a doubt, was an Aryan. The Pali Cannon describes the Buddha as tall, handsome, blue eyed...from Afghanistan to Northern India, you find people that could possibly be the descendants of the ancient Aryans...

Yes, he was an Aryan, as evidenced by language, - and blue eyes in the list give an important hint.
A mongoloid would have spoken Tibeto-Burman language.

Michael Witzel wrote on the origin of Sakyas:

Both the Malla and Vrji apparently immigrated into the east only after the end of the Vedic period, but well before the time of the Buddha (c. 400 B.C.). This must have been one of the last great infiltrations in Vedic times of western peoples into the lower Gangå area. More or less about this time the so-called second urbanization began as well.

Nevertheless, the settlement pattern in the east was not as homogenous as it was in the more western areas where the indigenous population had become Indo-Aryan in language and culture since the Mantra period. Instead, the Kosala-Videha area was one of great mixture of peoples. There were some earlier eastern Indo-Aryan settlers, the local Munda people and some Tibeto-Burmese elements. Then, various types and groups new immigrants entered from the areas further west. These were some brahmanically oriented tribes but also other non-orthoprax Indo-Aryan tribes such as the Malla and Vrji. They immigrated from northwestern India into Bihar which had been already settled by the old, para-Vedic Indo-Aryan tribes such as the Iksvåku, Kosala, Kåśi, and Videha.

Many of these tribes, including the Śakya to whom the Buddha belonged, are called asurya in ŚB. For it is the Sakya and their neighbors, the Malla, Vajji, etc. who are reported in the Påli texts as builders of high grave mounds, such as the one built for the Buddha. According to ŚB 12.8.1.5 the “easterners and others(!)” are reported to have round “demonic” graves, some of which may have been excavated at Lauriya in E. Nepal. These graves are similar to the kurgan type grave mounds of S. Russia and Central Asia. However, the origin of the Śakya is not as clear as that of the Malla and Vrji. They may very well have been (northern) Iranian, and would then constitute an earlier, apparently the first wave of the later Śaka invasions from Central Asia.

You're probably right. This man from Iran is probably what the ancient Persians looked like before the Arabs invaded... the ancient Persians and Northern Indians called themselves "Aryan".

"He, the Blessed One, is indeed the Noble Lord, the Perfectly Enlightened One;
He is impeccable in conduct and understanding, the Serene One, the Knower of the Worlds;
He trains perfectly those who wish to be trained; he is Teacher of gods and men; he is Awake and Holy. "
--------------------------------------------"The Dhamma is well-expounded by the Blessed One,
Apparent here and now, timeless, encouraging investigation,
Leading to liberation, to be experienced individually by the wise. "